.9999… bottles of beer in the wall, .9999… bottles of beer.
Take .0000…1 down and pass it around,
.9999…8 bottles of beer in the wall.
ETA: How soon before Little Billy in the back seat starts asking “Daddy, are we nearly there yet?”
“I told you, Billy, just practice counting all the reals between 0 and 1 and we’ll be there in no time!”
Is there a GQ medal for Brevity of OP Subject Header (BOSH)/Length of Thread (LOT)?
Moreover, because threads are always open this thread can be seen as a meta thread.
Or else the prize goes to any to any thread that scores 1. If we ignore locked threads.
I hope a mathematician can explain how zombies (that is, zombie threads) fit in. Actually, I’m sure this was discussed (!) above, but I’m comfortable with this. Plus it might help the mods should such a contest arise.
The appearance of the 8 means you made the sequence finite. You’re out of the game. We’re leaving you behind at the next rest stop.
I wish people would read the entire thread before posting! 
That the thread is a self-referential metaphor has already been noted:
.999 - Are we almost finished?
.9999 - Not yet, but you’re closer to your goal, closer to La petite mort
.99999 - When (gasp!) when is this [DEL]#%$&&#[/DEL] thread going to finish?
.999999 - I’m trying so hard…
.9999999 - Soon … soon …
.99999999 - Why must you torment me so?
If this prolongation continues, I worry that a
[SPOILER] VERY Not-safe-for-work
certain NSFW agency [/SPOILER]will start spamming this thread with ads.
Looks like it’s time for another
Have you read the entire thread first?
See my earlier post #771 for some thoughts apropos this.
Bingo.
But how does it fit in, exactly, that I did not read the full thread, but came in a few posts ago. (I actually glanced at this thread every day or so.)
Is that accounted for in your example?
Nonsense! Well, at most .0000…1% sensible anyway. Clearly I’m referring to the sequence of infinitely many 9 digits, followed finally by 8 in the last (∞’th) decimal place. Have not erik150x and Valmont314 convinced us all that it is meaningful to speak of the very last digit of an infinite sequence, specifically the one commonly (in this thread) written as .000…1 ?
And did I myself not propose a rigorous and sensible meaning for this notation a few posts back? In which, just as .999… turns out to be exactly 1, likewise .000…1 = exactly 0. (Hey, it’s just a notation with a proposed defined interpretation the simply supports what real mathematicians say about this, and tends to refute the infinitesimalists here. I never said it was a highly useful notation other than that.)
Well, you gotta take a week off work sometime to read the whole thread, at your convenience of course. (I myself got into it about the time erik150x was blathering about .999… not being equal to 1, but something less than that, but not much less. I think that was somewhere around post #100 or so.)
The posts by erik150x were just intelligible enough to stimulate some meaningful discussion. Once the real mathematicians began to give up on knocking some actual mathematical sense into him, however, they got off onto interesting tangents discussing alternative number systems, hyperreals, and stuff like that. (Tangents on which I myself, BTW, did not participate, those topics being rather over my head.)
With the more recent resurrection and Valmont314’s doctoral dissertations posted here, however, it seems we’ve all thrown serious discussion to the winds and are instead just posting silly fun responses.
Won’t you be pleased to join us? All together now . . .
.999…6 bottles of beer in the wall, .999…6 bottles of beer! . . .
Ok,
Well it appears that the discussion has died down to a nice serene level akin to that which one could find in a place of Worship. So what is written below should be read with the thought of it being spoken in a very gentle and quiet voice. Believe me I’m not here to piss anybody off.
I have read through most of the thread and there appears to be a lot of bickering and kicking around of snippets of mathematical concepts which seem to dance around the essence of the questions that need to be addressed but rarely address them directly. So here goes one more iteration (we have an infinite number at our disposal so one more is not going to hurt right?) in what will be my final attempt to pin this down once and for all here on this message board. I personally don’t feel the need to explore this issue any deeper than this so I will only comment from this point on to make small adjustments.
Ok here we go, let’s try to divide and conquer.
For starters the 800 some messages debating this equation can first be divided into those regarding first the practical reality of why the real number .999… is by definition equal to the real number 1 and the conceptual question of whether or not the concept .999… is identical to the concept 1.
The practical reality:
First, consider the following thought experiment which has symmetry with the real world. You are given a meter stick ruler which has markings of 0 meter, 1 meter, decimeters marked as .1 to .9, and centimeters indicated with marks but no numbers. You are asked to take a measurement which you can see falls at the 9th centimeter mark following the .9 decimeter mark. So you write down your measurement as .999 meters. Now you are asked to provide the results of the measurement using only numbers which were listed on the meter stick. The answer is .99 meters. You go back to your piece of paper and write down the fact that for all future measurements the points .999 and .99 which are clearly separated by an interval are in fact theoretically co-located. How does this relate to the real number system and the fact that the points .999… and 1 are co-located?
The precision of which a number can be represented in any system of measurement is directly and exactly related to the fundamental representation of minimum interval size in that system. For the decimal representation of real numbers it is abundantly obvious that there is a direct relationship between the number of decimal places in the smallest fundamental representation of interval given by .001 (for the case of 3 decimal places) and the number of decimal places allowed for representing precision. So if the number of decimal places was limited to three as above, the limits of precision in that system would be 3 decimal places and there would be no need for co-located points. Essentially in a balanced system, that is, a ruler which has every marking indicated with a number and from which one simply discards the interpolated part of the measurement that does not land on a marking, there are no co-located points. The relationship between the fundamental representation of interval in a system and the degree to which precision can be represented can be thought of as the water in a sink trap – it wants to be balanced in order for the system to work. For the case of the real number system, in order to eliminate the problem with the co-location of the points .999… and 1 one has to rigorously define the relationship between the number of decimal places allowed for .001 and the number of decimal places allowed for the precision with which numbers are represented. Again, in order for the system to be balanced (which I am defining as not having any co-located points), the number of decimal places in the least element of the set must be exactly equal to the number of allowed decimal places for representing precision. If this holds true and the system is balanced, one can further observe the fact that every member of the set can be represented as some multiple of the least element. If the system is not balanced then one cannot represent every member of the set as a multiple of the least element. In a balanced system the precision is a function of the least element , not the other way around. So, the bottom line is, if the ellipsis in the .999…represents an infinite number of nines, and .000…1 is not an element of the set, the system is not balanced and as a result .999… cannot be represented as a multiple of the least element of the set. The problem can be seen much more clearly if you simply consider the set of integers >9. Using this system for measurement, what is the minimum interval size that can be represented? 10 of course. It can’t be one because one is not a member of the set. So the answer to the question of “What is the difference between 13 and 12 ?” is very simply, either there is no difference, the difference has no representation therefore for the purposes of measurement the points are one and the same point and are thus co-located or the difference is 10. But they are not the same elements of the set and thus it cannot be said that 13 and 12 are identical. All of this, has its underpinnings in the fact that there is a very subtle attribute of the integers that we don’t ever even notice because it is so deeply ingrained: The interval size is always defined by the relationship of 1 to zero and is taken to be the reference for the difference between all successive integers. What does this mean in conceptual terms? It means that anytime we ask the question what is the difference between any two successive integers say 17 and 18 we are really asking the question of what is the difference between 1 and 0.
Now, consider the set (0,1). Could you ever use this system for making measurements? Any measurement made with such a system would have to be interpreted as either zero or one. If we operate on the reasonable rule that any measurement made must be finite every measurement made with such a system would be 1 and therefore we could deduce that points 1 and 0 are co-located yielding the nonsensical result that 1=0 (nonsensical if you interpret it to mean that when using this set to count things you can no longer distinguish between the existence of something or not). This implies that the system is not balanced. Why or how is it not balanced? Very simply the fundamental representation of interval that is being referenced by the interval is the interval itself. It appears that an additional condition for a system to be balanced is that there is at least one level of recursion of the fundamental representation of interval (which always involves the integer 1). For example this one level of recursion occurs in decimal representations of real numbers at the dot separator. So if we consider the system above with the additional elements from the first decimal place we have the set (0,1,.1,.2,.3,.4,.5,.6.,.7,.8,.9). There is a recursive relationship between 1 and .1 in that 1 can be represented exclusively using itself in the form of a multiple of .1. The least element is now .1 and can be referenced by 1 so that the system is balanced. It appears that in order for systems of measurement to be balanced the least element cannot equal the greatest element. Again, a balanced system of measurement must include at least one sub-level of intervals, much in the same way that in order to qualify as a ruler a ruler must have at least one level of markings on regular intervals.
So that’s my final answer to this entire conundrum:
Saying .999. = 1 is a very imprecise way of saying that the two distinct elements .999… and 1 of the set of Reals are co-located. This is due to the fact that the interval representation of the difference between the two members is not a member of the set, and the interval representation is not a member of the set because the system is not balanced, and the system is not balanced because the relationship between the least element of the set and the limits of precision is not rigorously defined…they are not identical any more than the elements 1 and 0 are identical in the example above.
Going deeper,
We can simply ask the question, what is the difference between 1 and zero (and 1 and every other integer) in terms of the conceptual characteristics of these integers
There are two interpretations to the answer just as there are two interpretations to the question. The question could be interpreted as what is the interval defined by the relationship between one and zero or it could be interpreted as what is the distinction between the two members of the set. The corresponding answers are 1 is referencing the interval and 1 is referencing the distinction between the two members of the set (0,1). Now, notice that the second question was not really answered. Pointing back to 1 does not explain the distinction, it simply points to the fact that one is the fundamental representation of self reference among the integers. The bottom line is, the integer 1 has very special properties which are distinct from other integers and are particularly relevant when integers are ever used for both counting AND measurement. Notice that asking the question of what the difference is between any two other integers does not result in the dual interpretation problem because it is clear that what is being referenced is the interval(of course the definition of this interval always points back to the interval (0,1) anyway).
The Conceptual Part:
The conceptual part of this argument is much simpler.
The simple question is…what is 1/∞?
Either it’s finite, zero, or both.
If it’s both, than clearly we are going back and forth arguing over what the correct interpretation of a Necker cube is, which is the logical equivalent of arguing in the real world over whether or not, based on our current understanding of Physics, a particle should be represented either as a particle or a wave…that is there is no correct single answer.
It amazes me, that a concept which was invented by the greatest genius in history, and whose use resulted in one of the most powerful tools in Physics, does not so much as have a Wikipedia page when Wikipedia has pages on every conceivable math topic ranging from a page for 5 to a page for Geometrodynamics.
My opinion of what 1/∞ is this:
It involves one so that is a giveaway that it contains the concept of self reference. Division by ∞ is in my opinion the fundamental representation of recursion. Much in the same way that 1 has the properties of quantity, interval, and self reference, 1/∞ has all of the properties of 1 with the included property of recursion (one level). Infinite recursion would thus have the fundamental representation of 1/∞ to the power of infinity.
Last but not least, “at infinity” (and this one goes out to you Exapno)
Running through some epsilon delta proofs you can show that if you make the following assumption:
0 < 1/∞ < 2/∞ < 3/∞ etc (up to 9)
You can see (notice I didn’t say prove) that 1/∞ is finite and can be seen as the epsilon interval in the lim 1/x as x goes to infinity = 0 limit evaluation.
That is you can derive a relationship between delta and epsilon to show that 1/∞ is finite.
Yes questionable but the above assumption seems to mesh with the behavior of the reals.
In conclusion, fill in the blank and answer this question, if you can think of a better way to formulate the question I am all ears!
The value of the function 1/x when evaluated ______________ is equal to 1/∞ which is:
a. Undefined
b. Finite
c. Zero
d. The answer points back to the question
e. b and c (a Necker cube)
f. b and d (a level of recursion of the representation of finite quantities)
g. c and d (a level of recursion of nothing)
h. b, c, and d (a recursive series of Necker cubes – ie nested Necker cubes)
nvm.
In your post, you acknowledge that the difference between .999… and 1 equals no real number. But this implies (if we assume 1 is real) that either .999 is not a real number, or .999 = 1. They cannot be distinct real numbers if their difference is not a real number. But you said they are both real numbers. Hence there is a contradiction in your current thinking.
Ok Frylock, for that gaff you are condemned to answering the question at the bottom.
There is no unique answer to the question. I need to know what axiom set you are using. If it is one in common use already, name it. If it is not, then define it.
Also, you seem to have put a blank in an odd place in your question… What purpose does that blank serve?
All ".999"s in that post should read “.999…”
Even the .999 meters?
Fill in the blank as you would if you were evaluating the value of a function for a finite value
Yes we all know that, there is a man standing behind you with a gun pointed at your head asking you for an intuitive answer…go for it.